Exponentials and one-parameter subgroups

Let $G$ be a Lie group with Lie algebra $\mathfrak g = T_eG$, and let $\exp:\mathfrak g\to G$ be the exponential map .

Lemma (Exponential–one-parameter subgroup).
For each $X\in\mathfrak g$, the map

is a smooth group homomorphism $(\mathbb R,+)\to G$, i.e. a one-parameter subgroup . Moreover,

Conversely, if $\gamma:\mathbb R\to G$ is a one-parameter subgroup, then there exists a unique $X\in\mathfrak g$ such that $\gamma(t)=\exp(tX)$ for all $t$; equivalently $X=\gamma'(0)$.

Context.
This lemma packages the correspondence between elements of $\mathfrak g$ and flows of left-invariant vector fields: the curve $\gamma_X$ is the integral curve through $e$ of the left-invariant field determined by $X$ (compare one-parameter subgroups as integral curves ). Locally, it is compatible with the fact that $\exp$ is a local diffeomorphism near 0 .